A new discrete dynamical system of signed integer partitions

Cattaneo, Gianpiero; Chiaselotti, Giampiero; Oliverio, Paolo Antonio; Stumbo, Fabio

doi:10.1016/j.ejc.2016.02.003

Cited by 30 publications

(5 citation statements)

References 34 publications

(3 reference statements)

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“…Network models are the natural way to formalize the evolution of any phenomenon that includes cells, gens, entities, or any kind of distinct elements which interact among them. Thus, it is not amazing the big amount of papers, published in the last decades, associated with sciences (see, for instance, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]) and engineering (see, for instance, [19][20][21][22][23][24]), where such models appear to represent different phenomena and study their behavior.…”

Section: Introductionmentioning

confidence: 99%

Predecessors Existence Problems and Gardens of Eden in Sequential Dynamical Systems

et al. 2019

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Section: Introductionmentioning

confidence: 99%

Predecessors Existence Problems and Gardens of Eden in Sequential Dynamical Systems

et al. 2019

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“…Now, since u ≡ A u − 1, it results that F S (u, a) = F S (u − 1, a). By (10), this implies that u ∈ J u,a . Therefore, by (9) and (19) we have that…”

Section: Representation Theorem For Closure Systemsmentioning

confidence: 85%

“…In this case we say that the pair (L, ψ) is a complemented involutory lattice. In general, the theory of the posets endowed of an involutory map is a well investigated research field [4,11,24,32], which also has links with complex analysis [7] and design theory [9,23] (moreover, for some studies concerning discrete dynamics on particular types of lattices endowed with involutory maps see [8,10]).…”

Section: Introductionmentioning

confidence: 99%

Lattice Representations with Set Partitions Induced by Pairings

Chiaselotti¹,

Gentile²,

Infusino³

2020

Electron. J. Combin.

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We call a quadruple $\mathcal{W}:=\langle F,U,\Omega,\Lambda \rangle$, where $U$ and $\Omega$ are two given non-empty finite sets, $\Lambda$ is a non-empty set and $F$ is a map having domain $U\times \Omega$ and codomain $\Lambda$, a pairing on $\Omega$. With this structure we associate a set operator $M_{\mathcal{W}}$ by means of which it is possible to define a preorder $\ge_{\mathcal{W}}$ on the power set $\mathcal{P}(\Omega)$ preserving set-theoretical union. The main results of our paper are two representation theorems. In the first theorem we show that for any finite lattice $\mathbb{L}$ there exist a finite set $\Omega_{\mathbb{L}}$ and a pairing $\mathcal{W}$ on $\Omega_\mathbb{L}$ such that the quotient of the preordered set $(\mathcal{P}(\Omega_\mathbb{L}), \ge_\mathcal{W})$ with respect to its symmetrization is a lattice that is order-isomorphic to $\mathbb{L}$. In the second result, we prove that when the lattice $\mathbb{L}$ is endowed with an order-reversing involutory map $\psi: L \to L$ such that $\psi(\hat 0_{\mathbb{L}})=\hat 1_{\mathbb{L}}$, $\psi(\hat 1_{\mathbb{L}})=\hat 0_{\mathbb{L}}$, $\psi(\alpha) \wedge \alpha=\hat 0_{\mathbb{L}}$ and $\psi(\alpha) \vee \alpha=\hat 1_{\mathbb{L}}$, there exist a finite set $\Omega_{\mathbb{L},\psi}$ and a pairing on it inducing a specific poset which is order-isomorphic to $\mathbb{L}$.

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“…A new type of symmetry geometry has been recently developed for some set operators and set systems induced by graphs [17,18], digraphs [14,28], information tables [12], some types of discrete dynamical systems [11,13,16] and metric spaces [17]. This geometry is based on a notion of generalized symmetry between c 2018 Australian Mathematical Publishing Association Inc. subsets of points (for metric spaces) or vertices (for graphs and digraphs), and we refer the reader to [17,18] for details concerning these notions and corresponding results.…”

Section: Introductionmentioning

confidence: 99%

Symmetry Geometry by Pairings

Chiaselotti

Gentile

Infusino

2018

J. Aust. Math. Soc.

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In this paper, we introduce asymmetry geometryfor all those mathematical structures which can be characterized by means of a generalization (which we call pairing) of a finite rectangular table. In more detail, let$\unicode[STIX]{x1D6FA}$be a given set. Apairing$\mathfrak{P}$on$\unicode[STIX]{x1D6FA}$is a triple$\mathfrak{P}:=(U,F,\unicode[STIX]{x1D6EC})$, where$U$and$\unicode[STIX]{x1D6EC}$are nonempty sets and$F:U\times \unicode[STIX]{x1D6FA}\rightarrow \unicode[STIX]{x1D6EC}$is a map having domain$U\times \unicode[STIX]{x1D6FA}$and codomain$\unicode[STIX]{x1D6EC}$. Through this notion, we introduce a local symmetry relation on$U$and a global symmetry relation on the power set${\mathcal{P}}(\unicode[STIX]{x1D6FA})$. Based on these two relations, we establish the basic properties of our symmetry geometry induced by$\mathfrak{P}$. The basic tool of our study is a closure operator$M_{\mathfrak{P}}$, by means of which (in the finite case) we can represent any closure operator. We relate the study of such a closure operator to several types of others set operators and set systems which refine the notion of an abstract simplicial complex.

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A new discrete dynamical system of signed integer partitions

Cited by 30 publications

References 34 publications

Predecessors Existence Problems and Gardens of Eden in Sequential Dynamical Systems

Predecessors Existence Problems and Gardens of Eden in Sequential Dynamical Systems

Lattice Representations with Set Partitions Induced by Pairings

Symmetry Geometry by Pairings

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